Fire Prince Ozai, his wife Ursa, his son Crown Prince Zuko, and his daughter Princess Azula.Shown here at their beach house on Ember Island two months before Azulon died and Ursa Vanished.

IMO All types of ponies can bend since they use their limbs to use their element. So there can be non-bending unicorns that can still manipulate things or pegasi who can shoot fire balls. That is my thoughts I have on it that I have had for a long time. And by long time I mean forty five seconds ago.

Here's my theory for the species, Avatar is special, when found, he/she will grow a wing or horn to become an alicorn. Fire is Unicorn, Water is pegasus & unicorn (unicorn has extra talent for healing, if not bloodbending), air is wings & earth is obviously earth ponies.

So in this case, it's recommended that you give the horn, Ozai needs it strong enough for using it during Sozin's comet, if you're going to fix it.